an analysis of covid-19 in europe based on fractal

$: An analysis of Covid-19 in Europe based on fractal$
Article

An analysis of Covid-19 in Europe based on fractal dimensionand meteorological data

Cristina Maria Pacurar 1,†,‡* , Victor Dan Pacurar 2,‡ and Marius Paun 2,‡

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Citation: Pacurar, C.M.; Pacurar,

V.D.; Paun, M. An analysis of

Covid-19 in Europe based on fractal

dimension and meteorological data.

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1 Faculty of Mathematics and Computer Science, Transilvania University of Brasov, 50 Iuliu Maniu Bld., email:[email protected]

2 Faculty of Silviculture and Forest Engineering, Transilvania University of Brasov, 1 S, irul Beethoven Street* Correspondence: [email protected]‡ These authors contributed equally to this work.

Abstract: The present paper proposes a fractal analysis of the Covid-19 dynamics in 45 Europeancountries. We introduce a new idea of using the box-counting dimension of the epidemiologic curvesas a means of classifying the Covid-19 pandemic in the countries taken into consideration. Theclassification can be a useful tool in deciding upon the quality and accuracy of the data available.We also investigated the reproduction rate, which proves to have significant fractal features, thusenabling another perspective on this epidemic characteristic. Moreover, we studied the correlationbetween two meteorological parameters: global radiation and daily mean temperature and twoCovid-19 indicators: daily new cases and reproduction rate. The fractal dimension differencesbetween the analysed time series graphs could represent a preliminary analysis criterion, increasingresearch efficiency. Daily global radiation was found to be stronger linked with Covid-19 new casesthan air temperature (with a greater correlation coefficient -0.386, as compared with -0.318), andconsequently it is recommended as the first-choice meteorological variable for prediction models.

Keywords: Covid-19, fractal analysis, box-counting dimension, epidemic curves, reproduction rate,global radiation, daily mean temperature

1. Introduction

Nowadays, the importance of thoroughly studying epidemics has been clearly em-phasised. Thus, acquiring information about a pandemic from every possible source canprove to be crucial in managing not only the current pandemic, but also future epidemics.Extracting new information from the available data is of utmost importance at presenttimes, as the current pandemic seems far from ending, even after more than a year anda half since it started. In this respect, the first part of the present paper aims to analysethe epidemic curves corresponding to Covid-19 pandemic as fractals and determine theircomplexity based on the fractal dimension associated, which has not yet been done, as faras we know.

The central attention received by the current Covid-19 pandemic has a significant echoin almost every area of research. Researchers from every field have joined forces to discoveras much as it can be unrevealed about the SARS-CoV-2 virus. As far as Mathematics isconcerned, Covid-19 has been thoroughly studied. Many researchers tried to model thevirus spread (see, for example [1–8], and many more). For a literature review on the subjectsee also [9–13]. As far as fractality is concerned, analysing the current pandemic from afractal point of view has also been tackled in different research studies (see [14–20]).

Among the numerous research directions related to Covid-19, assessing the correlationbetween the new cases of Covid-19 registered and meteorological data has been in thespotlight. From the numerous researches related to this topic we mention the followingones along with all their bibliographic references: the systematic review of Majumder andRay on the correlation between meteorological data and Covid-19 (see [21]), the systematicreview of Briz-Redon and Serrano-Aroca (see [22]), the survey of Chen et al. on the relationbetween climate and the current pandemic (see [23]), and many others (see [24–26]).

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 September 2021 doi:10.20944/preprints202109.0425.v1

© 2021 by the author(s). Distributed under a Creative Commons CC BY license.

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The present study is aimed to contribute at the improvement of the forecasting basisfor the Covid 19 pandemics evolution, by using fractal analysis in two main directions:assessing the quality of the reported data and the effect of weather conditions on the diseaseincidence and reproduction rate.

On one hand, the Covid-19 epidemics is analysed from a fractal perspective, em-phasizing the fractal structure of the epidemiologic curves. The fractality of the Covid-19epidemiological curves has been argued in previous studies (see [20]). However, as far aswe know, analysing the complexity of Covid-19 through the fractal dimension of epidemio-logic curves has not been done before.

One of the immediate benefits of comparing the fractal dimensions of the epidemi-ologic curves in different countries gives us the possibility to discover the places wherecommunicating Covid-19 related data has been more accurate and where testing in orderto discover the active Covid-19 cases has been most consistent. Moreover, the analysis in acomparative manner of the complexity of epidemiological curves in European countriescan help discover where the measures against the pandemic have been the most efficient.In order to better evaluate the latter, we have also studied the reproduction rates in thecountries that we considered.

On the other hand, we investigated the influence of the meteorological conditions onthe Covid-19 evolution parameters. Many airborne spreading diseases are well known ashighly dependent on weather, with an obvious seasonality in mid-latitudes regions, withinfluenza as a typical example, and the new Corona virus infection making no exception.Although analysing the correlations between Covid-19 and meteorological data is not newand has been the subject of several interesting research papers, our analysis reveals somenew possible correlations representing an interesting novelty. There were selected twometeorological variables, temperature, which was considered in a very large number ofstudies and confirmed to be well correlated with Covid progress, and solar radiation, whichwas seldom taken into study, despite its role as a primary energy source for atmosphericprocesses, thus decisively affecting temperature regime, and additionally having a directinfluence on the virus spread, due to its sanitising action, given the lethal effect of the UVspectral component for viruses.

2. Materials and methods

The present paper studies from a fractal perspective the evolution of Covid-19 in theEuropean countries which have publicly shared Covid-19-related data. All data related toCovid-19 (new cases, reproduction rates, vaccinations) is collected from Our World in Data(see [28], [27]). The analysed data is composed of entries from the first day when Covid-19cases were recorded in every country, until 28 July 2021. In Appendix A (in Table A1), it ismarked the first day when Covid-19 cases were recorded in each of the countries that weconsidered.

The climatic data were extracted from the website of the European Climate Assessment& Dataset project (see [29], [30]). We analysed data corresponding to a one-year-period,from June 1st 2020 to May 31st 2021.

We investigated the data from a fractal point of view using the fractal dimension. Frac-tal dimension is a measure of the complexity of an object. There are numerous variants ofthe fractal dimension. Among them, one of the dimensions which is most used in practiceis the box-counting dimension (or Minkowski–Bouligand dimension, or Minkowski di-mension, see [31]). Throughout the article, by fractal dimension we mean the box-countingdimension.

Although box-counting dimension is not usually the most suitable dimension foranalysing time series data, in the present case, for analysing the complexity of epidemi-ologic curves related to Covid-19, has been proven to be the most appropriate. The factthat the box-counting dimension usually measures the space-filling properties of a structure,although it can be an impediment in most cases when analysing time-series data (see [32]),in the present case, the aspect ratio choice (which is scaled relative to the highest number



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of Covid-19 cases recorded) does not alter the results used for comparison purpose. This isdue to the fact that analysing the complexity of the epidemiologic curves must be scaledaccording to the population of the country. Moreover, the chosen scaling is also relevantfor understanding which countries have managed better the Covid-19 crisis.

We used Python programming language to compute the box-counting dimension andwe adapted the code developed by Rougier [33].

The correlation represents a statistical feature of two random variables or bivariate datawhich measures the relationship between the latter. In the current paper, by correlation weunderstand the degree in which two datasets are linearly related. Correlation is a statistichighly used in practice. The correlation coefficient is the measure which determines thecorrelation, i.e. the degree to which variables move in relation to each other. For the currentstudy we use the Pearson correlation coefficient (see [34]) which returns values between -1and 1.

For analysing the correlations between the sets of data that we considered, we haveused the correlation feature (dataframe.corr()) which can be found in the Python packagepandas (see [36]). The correlation function finds the pairwise correlation of the columns inthe dataset (for more information about the function see [37]).

As a case study, we chose to analyse the correlations between Covid-19 data andmeteorological data for Switzerland. As Covid-19 data is reported for the entire country,and the meteorological data are reported from stations in particular locations, we com-puted the mean value for the meteorological data considered between three stations inSwitzerland, situated in different parts of the country. The chosen stations are Zurich(Zuerich/Fluntern, Latitude: +47:22:59, Longitude: +008:34:00, Elevation: 555m), Geneva(Geneve Cointrin, Latitude: +46:15:00, Longitude: +006:07:59, Elevation: 420m) and Lugano(Latitude: +46:00:00, Longitude: +008:58:00, Elevation: 273m).

The significance and the confidence level of the correlations were determined byusing the F (Fisher probability distribution) test, using the following procedure. For eachcorrelation coefficient (r) the corresponding value of F was calculated with the formula inequation (1)

F =r2(N − 2)

1 − r2 (1)

where N − 2 is the number of freedom degrees, equal to 363, given the sample size ofone year (365 days). These calculated values were compared with three thresholds, corre-sponding to the theoretical F values, for 1 and 363 freedom degrees and a transgressionprobability of 5%, 1% and 0.1%, equal to 3.87, 6.71 respectively 11.01. Calculated valuesgreater than these three thresholds are indicating a significant, distinctively significantrespectively a very significant correlation.

3. Results3.1. Fractal dimension of Covid-19 epidemiologic curves

We chose to analyse data related to Covid-19 for European countries. We computedthe graphs of the new cases registered daily in a bidimensional canvas where the transver-sal axis corresponds to the date of the new cases recorded on the longitudinal axis, thusobtaining the epidemiologic curves corresponding to each of the countries that we consid-ered.

We determined the fractal dimension for 45 European countries. The box-countingdimensions computed are shown in Appendix B (see Table A2). We represent the box-counting dimensions obtained, sorted in ascending order in Figure 1.

The results obtained for the fractal dimension corresponding to each country can giveus an inside on the accuracy of the data reported. The highest fractal dimension computedis for Sweden, a dimension of 1.4732, and the smallest fractal dimension is recorded forVatican, a dimension of 1.2912. However, the two extreme data are both a result of aninaccuracy of the data provided. The main reason for a fractal dimension too close to 1 is



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Figure 1. Box-counting dimensions computed for the 45 Europen countries analysed

the inclusion in the analysis of really small countries, which usually reported less casesof Covid-19 (less than a maximum of 500 cases reported in a day). In this situation wecan exclude the following countries from further analysis for the purpose of the presentresearch: Andorra (maximum of 299 cases reported in a day), Iceland (maximum of 126cases reported in a day), Liechtenstein (maximum of 62 cases reported in a day), Monaco(maximum of 45 cases reported in a day), San Marino (maximum of 89 cases reported in aday), Vatican (maximum of 7 cases reported in a day).

For further investigations on the fractal dimension we can split the remaining 39countries into three main categories, as in Table 1

Category Dimension interval Accuracy of data- [1,1.3) IrrelevantA [1.3,1.38) Highly accurateB [1.38, 1.45] Mostly accurateC (1.45,2] Noisy and mostly inaccurate

Table 1: Classification of datasets based on fractal dimension

Classifying the countries based on the box-counting dimension as in Table 1, we obtaina clearer image upon the data sets which can be further taken into consideration. In Figure2 we depict the classification of the countries based on the box-counting dimension on thegraph of the dimensions of the 39 countries sorted in ascending order. Analysing AppendixB (Table A2) and Table 1 we can observe that from the 39 countries considered, there is nocountry with irrelevant data, there are 9 countries with highly accurate data, 26 countrieswith mostly accurate data and 4 countries with noisy and mostly inaccurate data.

In order to better visualise the relevance of classifying the countries into three cate-gories, we present an example of graphs corresponding to a country from each category, asfollows: category A - Figure 3, category B - Figure 4 and category C - Figure 5.

Analysing Figures 3, 4 and 5, we can clearly see that the data in Figure 3, correspondingto Russia, are highly accurate and more frequently reported as the variations are realisticand exact to the evolution of the disease. For Figure 3, the box-counting dimension is 1.31.In Figure 4, the data is more chaotic, and although Covid-19 has some chaotic features,



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Figure 2. Classification of countries based on box-counting dimension

Figure 3. Graphs of new cases registered inRussia (A)

Figure 4. Graphs of new cases registered in theNetherlands (B)

Figure 5. Graphs of new cases registered inSweden (C)

most of the high variations are actually noise generated by the channels and frequencychosen for reporting the data. The box-counting dimension for Figure 4 is 1.39. Even so, thedata reported for the countries in category B are accurate and can be used for predictionand analysis. In Figure 5, the situation is extreme, as the data are clearly too noisy to be



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analysed and must be subject to serious filtering and corrections before being ready foranalysis. The box-counting dimension computed for Sweden is 1.47.

3.2. Fractal patterns of reproduction rates

An essential parameter of an epidemic is the reproduction rate (or reproduction num-ber) which indicates the number of new cases that will be generated by each new infectionamong the healthy individuals. Studying the reproduction rate can indicate whether theepidemic is expanding (a higher reproduction rate), or it is likely to disappear (a lowerreproduction number). The evolution of the Covid-19 pandemic, which is characterised bythe constant appearance of new waves, seems to be governed by periodic functions, whichposses, besides a great deal of similarity, an even greater amount of fractality.

Figure 6. Graph of reproduction rate inDenmark

Figure 7. Graph of reproduction rate inRomania

Figure 8. Graph of reproduction rate inSwitzerland

Plotting the reproduction rates may pinpoint some interesting fractal features ofthe reproduction rate. We can observe fractal patterns emerging from the graphs of thereproduction rates in most of the countries analysed.

Analysing the three example graphs from Figures 6, 7 and 8 we can observe theresemblance of the graphs of the reproduction rates with a classical fractal, the randomBlancmange curve (or Takagi curve) which is depicted in Figure 9. Splitting the graphs inFigures 6, 7 and 8 and looking separately at smaller parts of the graph, we can better seethe resemblance. As an example, see Figure 10. Thus, treating the reproduction rate as afractal (having a fractal behaviour), might be an appropriate way to a better analysis of it.

3.3. Meteorological data and Covid-19

In the evolution of airborne spreading viral diseases, such as COVID-19 (the infectionwith the SARS-CoV-2 coronavirus, which is mainly disseminated through aerosolizeddroplets) the weather conditions play an important role. Their influence is both direct,affecting virus survival time and human host vulnerability, and indirect since weatheraffects people behaviour, changing the interactions frequency etc. Obviously, the factorsdeterminant for people conducts, especially in a pandemic context, are numerous, and



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Figure 9. Modified random Blancmange curve

Figure 10. A browsed window showing fractal features of the graph of reproduction rate for Denmark

quite often the non-meteorological issues, primarily social circumstances, are causing thenew "waves" occurrence and dynamics.

For this study we selected two meteorological parameters, daily mean values of airtemperature and solar radiation, and two parameters describing Covid-19 dynamics: dailynew cases and reproduction rate. Temperature was chosen because it was intensivelystudied in this pandemic context, with numerous articles reporting a significant correlationwith Covid-19 evolution, thus offering an extensive base for results discussion. Therewere many other weather data considered (relative humidity, rainfall amounts and windcharacteristics etc.), but almost never solar radiation. This could directly impact the virusspread, due to its lethal effect on the germ, especially of the ultraviolet component ofincoming sun-rays, on both contaminated surfaces and through aerosol droplets. Thus, wedecided to also investigate the correlation between solar radiation data and pandemicsevolution parameters, for the previously mentioned unique direct effect, but also for anadditional reason: solar radiation is the main climate genetic factor, the primary energysource for atmospheric processes, consequently a good candidate for a meteorologicalindependent variable in a Covid-19 forecasting model. This is very important because allweather data are well correlated and this autocorrelation could bring stability issues formodels, with internal regression equations involving multiple meteorological variables.

As regards the Covid-19 parameters, the selection was similar, including the numberof daily new cases, which was largely considered in researches focusing weather andCovid-19 interactions, and the reproduction rate, seldom taken into account in such studies,probably being considered mainly determined by social behaviour.



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Taking into account data availability and quality, we chose to test the correlationbetween meteorological data and Covid-19 data for Switzerland. Since the Covid-19 relateddata is available only on a larger area, in most cases the whole country, we had to processthe meteorological data to get an overall image of the area under study. For assessingthe meteorological parameters representative for the entire country, we computed a meanvalue. For Switzerland, we computed the mean daily data, as averages of the data reportedby three stations: Zurich, situated in the North of the country, Geneva, in the Southwestand Lugano, located in the Southern part of Switzerland. As concerns solar radiation, theSwiss data used were daily values of global solar radiation (the sum of direct beam anddiffuse radiation).

For analysing the correlations between the two pairs of datasets (new cases and repro-duction rate, air temperature and global radiation), the calculated correlation coefficientsare synthesised in the following table (Table 2).

New cases Reproduction rateMean temperature -0.318** 0.273**Global radiation -0.386** 0.097

Table 2: Correlations between datasets for Switzerland**highly significant data

The significance of the correlations was established using the procedure describedin Section 2 (Materials and methods). The values of the F statistics, corresponding to thecorrelation coefficients in Table 2, calculated with formula (1), are presented in Table 3.

New cases reproduction rateMean temperature 40.44** 29.23**Global radiation 63.56** 3.45

Table 3: Calculated F values, used for testing the correlation significance**highly significant data

The values marked with double asterisk indicate a very high significance, all threebeing greater than the 11.01 threshold (corresponding to a 0.1% transgression probability).The F value, associated with the low value (0.097) of the correlation coefficient betweenglobal radiation and reproduction rate is lower than 3.87 (corresponding to 95% confidencelevel) which indicates no significant correlation.

It is noteworthy that, despite the very strong negative correlation with the numberof new cases, determined for both mean temperature and global radiation, the second isconsiderably more intense, as the F statistics suggests. The significant positive correlationbetween reproduction rate and mean air temperature, possibly indicates the association ofcold weather with lower humans’ mobility, while warmer conditions could increase socialinteractions.

In addition to this classical approach, the graphical representations of the studieddatasets (see Figure 11 and Figure 12), were fractally analysed, their box-counting dimen-sions could be read in Table 4.

Datasets Fractal dimensionMean temperature 1.3866Global radiation 1.4829Daily new cases 1.4364

Reproduction rate 1.2897

Table 4: Fractal dimensions of the graphical representation of the study time series



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Figure 11. Charts of daily air temperatures and global radiation Swiss time series, for the studyperiod

Figure 12. Graphs of daily new cases and reproduction rate values time series

The values of the fractal dimension are clearly consistent with the visual evaluation ofthe graphs’ complexity, as can be seen in Figures 11 and 12, with an obvious neater shapefor the reproduction rate curve.

For comparing these study datasets graphs, we calculated the differences betweentheir fractal dimensions, obtaining the values in Table 5.

Daily data New cases Reproduction rateMean temperature 0.049 0.097Global radiation 0.047 0.193

Table 5: Differences between the fractal dimensions of the study datasets graphs

Comparing these pairwise differences with the correlation coefficients, we could ob-serve a clear similarity. Low differences (3-7% of the average fractal dimension) indicate agood association of the corresponding graphs, thus a correlation between datasets, whilea higher difference (approximately 14% of the mean value of the four fractal dimension)suggests the lack of correlation between datasets. A great difference indicates correlationabsence and the pair of values could be excluded from further investigations. The recip-rocal is not complying. A low difference suggests a possible correlation, which could beconfirmed or not by subsequent testing.

4. Discussions

The obtained results related to the box-counting dimension of the new cases of Covid-19 recorded in the 39 analysed European countries gives us an insight on the accuracy ofthe data reported. The importance of gathering such information is crucial for determiningwhat kind of prediction model is suitable for each country and whether the availabledata might be an efficient training dataset for a prediction model. The difference betweenanalysing the box-counting dimension and other means of measuring the complexity of astructure is that fractal dimension can give us an insight on the complexity of the structureas well as pointing to the noise which is interfering with the data.



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Our classification based on fractal dimension of the analysed data allows the exclusionof some countries, where the data might interfere with the accuracy of the prediction. Thus,besides offering an insight on the complexity of the data, the method which we proposeaims to facilitate the data selection step and helps obtain more accurate prediction models.

The highly significant negative correlation (-0.318) between mean air temperature andnew Covid-19 cases that we found in this study (with a temperature range from -6.4oC to23.7oC, in the study period), integrates in the mainstream of similar research results.

There are many studies, undertaken in different climate regions of the world, stating anegative strong correlation between these daily parameters. A worldwide study, focussingon 166 countries (see [38]), in climates with temperatures ranging from -5.3oC to 34.3oC, found that air temperature and relative humidity were negatively correlated withdaily cases, the number of new cases decreasing with 3.1% for temperature increasing1oC. Similar results were obtained in a tropical, warm climate (temperatures between16.8oC and 27.4oC, in Brazil, where the reduction rate was estimated to 4.9% , for a unitincrease of temperatures (see [39]). A considerably higher correlation, with a decrease ofnew cases reaching 13.53% for a 1oC temperature increment (see [40]), was found in Africa(temperatures in the study samples spanning from -2.4oC to 33.4oC). In Jakarta, Indonesia,also in a warm climate region, the average daily temperatures (from 26.1oC to 28.6oC) weresignificantly correlated with COVID-19 cases (correlation coefficient equal to 0.392), whileother weather data (Tmin, Tmax, humidity and rainfall) occurred to be not correlated (see[41]).

A study (see [42]), focusing on the initial period of the pandemics (from December2019 to March 2020) for all countries (including China) stated a significant, negative,correlation, between daily air temperatures and COVID-19 incidence (temperature range-33.9oC to 34.3oC). The significant effect of daily temperatures (ranging from -22oC to 26oC)on disease incidence, was also documented in China (see [43]), in New York State, USA, fora temperature interval between -3.4oC and 25oC (see [44]) and in Spain (see [45]), wherethe temperatures interval, in the study period was [1oC, 23.2oC.].

The relationships between air temperatures and new Covid-19 cases occurrence arefar from being fully understood. There is a large number of research outcomes, some ofthem mentioned above, indicating their significant negative correlation, but there are alsostudies reporting the opposite. For instance, a study undertaken in the Norwegian capital(see [46]), for a period with air temperature spanning from -0.5oC to 21.9oC, reportedpositive correlations between daily new cases and both maximum and mean temperatures(correlation coefficients of 0.347 and respectively 0.293). There is also a Chinese study (see[47]), analysing 122 Chinese cities, documenting a 4.86% increase in COVID-19 cases foreach 1oC temperature increase, but only in cold periods, with temperatures below 3oC.

The other meteorological variable considered in this study, global radiation, was foundto be stronger correlated with Covid-19 new cases than air temperature (with a greatercorrelation coefficient -0.386, as compared to -0.318, and the difference is emphasised by theassociated F values, 63.56 and 40.44 respectively). As mentioned before solar radiation wasseldom taken into survey, but a study (see [48]), in Rio de Janeiro, Brazil, for two months inthe spring of 2020, found similar results: a very significant correlation of both temperatureand solar radiation with Covid-19 incidence, but considerably stronger for the latter (witha correlation coefficient, -0.609, greater than -0.406, calculated for air temperature).

5. Conclusions

The present paper is composed of two main parts which cannot be separated. Themain novelty of the first part is computing the fractal dimension of the epidemiologiccurves in order to obtain new characteristics of the complexity of the pandemic in thecountries in Europe. Moreover, the analysis of the fractal features of the reproduction ratesis also a new idea brought by the present paper.

On one hand, the classification of the epidemic spread in different countries based onthe fractal dimension of the epidemiologic curves that we propose can be an significant



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tool in selecting the datasets which can be used for modelling the pandemic and predict itsbehaviour. The accuracy of the data, which is of utmost importance, is distinctively trackedby the fractal dimension of the epidemiologic curves. On the other hand, the analysisof the reproduction rate, which we propose, revealed the existence of noticeable fractalpatterns. Treating the reproduction rate as a fractal can further prove to be an efficient toolin discovering more much needed information not only about the Covid-19 pandemic, butalso about other types of contagious diseases.

Moreover, the fractal dimension differences between the graphical representations ofmeteorological data and Covid-19 daily progress parameters time series, could represent apreliminary analysis criterion, enabling the selection of potentially significantly correlatedvariable pairs (actually, eliminating the improbable candidates), thus increasing researchefficiency.

Weather conditions influence COVID-19 dynamics (both directly, affecting virus sur-vival time and human host vulnerability, and indirectly, impacting people behaviour). Asregards the two meteorological variables considered herein, daily global radiation, wasfound to be stronger linked with Covid-19 new cases than air temperature (with a greatercorrelation coefficient -0.386, as compared with -0.318). Given this stronger correlation,we recommend daily solar radiation amounts as the first choice for pandemics dynamicsforecasting models, especially for those using a single meteorological variable, in order toincrease stability (avoiding autocorrelation issues). Solar radiation, the primary energysource for atmospheric processes, affecting decisively all other weather elements (includingair temperature), has also a direct influence on the virus spread, due to its sanitising action,given the lethal effect of the UV spectral component for viruses.

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Author Contributions: Conceptualization, C.M.P. and V.D.P.; methodology, C.M.P. and V.D.P.; soft-ware, C.M.P.; validation, C.M.P., V.D.P. and M.P.; formal analysis, C.M.P. and V.D.P.; investigation,C.M.P. and V.D.P.; resources, C.M.P., V.D.P. and M.P.; data curation, V.D.P. and M.P.; writing—originaldraft preparation, C.M.P. and V.D.P.; writing—review and editing, C.M.P, V.D.P. and M.P.; visual-ization, C.M.P.; supervision, V.D.P.; project administration, C.M.P.; funding acquisition, C.M.P. Allauthors have read and agreed to the published version of the manuscript.

Funding: The APC was funded by Transilvania University of Brasov

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.MDPI Multidisciplinary Digital Publishing InstituteDOAJ Directory of open access journalsTLA Three letter acronymLD Linear dichroism



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Appendix A

Country First case Country First casereported reported

Albania 3/9/2020 Luxembourg 2/29/2020Andorra 3/2/2020 Malta 3/7/2020Austria 2/25/2020 Moldova 3/8/2020Belarus 2/28/2020 Monaco 2/29/2020Belgium 2/4/2020 Montenegro 3/17/2020Bosnia and Herzegovina 3/5/2020 Netherlands 2/27/2020Bulgaria 3/8/2020 North Macedonia 2/26/2020Croatia 2/25/2020 Norway 2/26/2020Cyprus 3/9/2020 Poland 3/4/2020Czechia 3/1/2020 Portugal 3/2/2020Denmark 2/27/2020 Romania 2/26/2020Estonia 2/27/2020 Russia 1/31/2020Finland 1/29/2020 San Marino 2/27/2020France 1/24/2020 Serbia 3/6/2020Germany 1/27/2020 Slovakia 3/6/2020Greece 2/26/2020 Slovenia 3/5/2020Hungary 3/4/2020 Spain 2/1/2020Iceland 2/28/2020 Sweden 2/1/2020Ireland 2/29/2020 Switzerland 2/25/2020Italy 1/31/2020 Ukraine 3/3/2020Latvia 3/2/2020 United Kingdom 1/31/2020Liechtenstein 3/4/2020 Vatican 3/6/2020Lithuania 2/29/2020

Table A1. First day of Covid-19 cases reported per country



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Appendix B

Country Dimension Country DimensionAlbania 1.3547 Luxembourg 1.4344Andorra 1.4042 Malta 1.3596Austria 1.3467 Moldova 1.4661Belarus 1.4393 Monaco 1.4330Belgium 1.3638 Montenegro 1.3967Bosnia and Herzegovina 1.4368 Netherlands 1.3918Bulgaria 1.4225 North Macedonia 1.4514Croatia 1.4442 Norway 1.4065Cyprus 1.3945 Poland 1.4056Czechia 1.4670 Portugal 1.3811Denmark 1.3630 Romania 1.4185Estonia 1.3864 Russia 1.3112Finland 1.4263 San Marino 1.4612France 1.4068 Serbia 1.3564Germany 1.4589 Slovakia 1.4367Greece 1.4398 Slovenia 1.4618Hungary 1.4006 Spain 1.4497Iceland 1.4020 Sweden 1.4732Ireland 1.3096 Switzerland 1.4155Italy 1.395 Ukraine 1.4256Latvia 1.4471 United Kingdom 1.3476Liechtenstein 1.4064 Vatican 1.2912Lithuania 1.4036

Table A2. Box-counting dimension computed for daily new Covid-19 cases registered in each country



an analysis of covid-19 in europe based on fractal

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